Non-uniform weights chairman assignment
Determine whether, for fractional assignments x ∈ [0,1]^{m×n} and nonuniform weights d_{ij} > 0 depending on both job j and machine i, there exists an integral assignment y ∈ {0,1}^{m×n} such that, for all rows i ∈ [m] and all prefixes t ∈ [n], the weighted prefix discrepancy satisfies |∑_{j=1}^t d_{ij}(x_{ij} − y_{ij})| ≤ max_{i ∈ [m], j ∈ [n]} d_{ij}.
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We ask for a version with non-uniform weights which may shed light on the unrelated machines scheduling, where job j has processing time d_{ij} on machine i: Let m, n be positive integers and d \in R{m \times n}{>0}. For every fractional assignment x \in [0,1]{m \times n}, there is an assignment y \in {0,1}{m \times n} so that for every i\in[m] and t\in[n], \Big|\sum{j \in [t]} d_{ij} (x_{ij}-y_{ij}) \Big| \le \max_{i\in [m], j \in [n]} d_{ij}.